File talk:Ensemble quantum 1DOF all states.png

Not eigenstates of Hamiltonian?

Dear Nanite, thank you for your contribution to your nice gallery. I am however concerned about the correctness of the states shown in the figure (or, more precisely, amplitudes of probability thereof).

As plotted, they seem NOT to be eigenstates of Hamiltonian -- weak coupling between two harmonic oscillators should lead to line splitting. This splitting is visible nowhere on the set of the states, yet it should be very strong and apparent for the energies close to the local energy maximum. Instead, each potential well seems to be filled with its own independent states.

I please you to review the computational routine you used to generate data for this otherwise excellent image.

With kind regards, FDominec (talk) 17:25, 29 December 2015 (UTC)

@FDominec: Ah sorry, I missed your comment! Well, better late than never.

In this case I set up the Hamiltonian as a matrix on finite difference spatial points, and just looked for eigenvectors. So it should be correct. As you say some splitting/tunneling should occur, but I guess in this case it is just too small. I was also surprised that none of the eigenstates were hybridized and delocalized over both wells.

But I think the explanation is thus: because the two wells are not identical, there are only some accidental near-degeneracies. And since the tunnel coupling is so weak, the states remain essentially non-hybridized. Or perhaps they do hybridize, but the splitting is too small compared to the smearing I added.

Certainly I have to say, carrying out the exercise of making this figure challenged some of my assumptions about quantum mechanics, and its relationship to classical mechanics.

Cheers, --Nanite (talk) 15:09, 26 April 2016 (UTC)

@Nanite: Today I have written a simple Python script for finding the 1D eigenstates, available here: https://gist.github.com/FilipDominec/9ab21b7eecb1875fcb2eba9958791cc1 I must acknowledge that your computations were correct, and my expectations were somewhat unrealistic. So I am glad to have taken the same exercise myself! If interested, please take a brief look at the bottom of the referred github "gist", where I give some comments to the numerical results. Regards, FDominec (talk) 14:49, 2 November 2016 (UTC)

@FDominec: Nice :) One more thing I remember now, I was also playing around with the 'hbar' value in my code to make things more or less quantum. By using a large hbar, one gets fewer states and more tunneling, for a given potential well. Nanite (talk) 19:02, 2 November 2016 (UTC)

@Nanite: Good point. I think that what you write about "more or less quantum" can be physically better realized by just choosing a different (quasi)particle mass. In practice, one can simulate (and fabricate) a 1-D semiconductor quantum well shallow enough to confine heavy holes only, but no light holes. FDominec (talk) 09:34, 6 December 2016 (UTC)